# Efficient Wireless Charger Deployment for Wireless Rechargeable Sensor Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

^{2}–1 μW/cm

^{2}and low radio frequency-direct current (RF-DC) conversion values of 10%–30%, a larger amount of total available power can be harvested by utilizing high gain antennas. Some antennas designed by the authors were shown [3]. For example, a folded dual-bad antenna at 915 MHz and 2.45 GHz was designed to harvest RF power from cellular and WiFi sources with power densities about 1 μW/cm

^{2}. Moreover, a folded dipole antenna was designed to harvest RF power from a two-way radio at the UHF band (464 and 468 MHz). They also implemented a prototype of an embedded microcontroller-enabled sensor platform successfully powered by an ambient UHF digital TV signal (512–566 MHz) from a broadcasting antenna that is 6.3 km away.

## 3. Modeling and Problem Definition

^{2}, that is, ${P}_{\mathrm{r}}=\mathsf{\alpha}{D}^{-2}$, where α is a constant and D is the distance between the charger and the harvester. This paper utilizes a more general empirical model ${P}_{\mathrm{r}}=\mathsf{\alpha}{D}^{-\mathsf{\beta}}$, where β > 0, to estimate the energy received by the harvester. Later in Section 5, we will show how to apply the power regression analysis to derive appropriate α and β values for the model to fit experimental data accurately.

## 4. Proposed Algorithms

_{x}and o

_{y}be the projection points of $\stackrel{\rightharpoonup}{g{s}_{{a}_{x}}}$ and $\stackrel{\rightharpoonup}{g{s}_{{a}_{y}}}$ on the unit sphere O centered at grid point $g$, and let d

_{xy}be the Euclidean distance between the two points. There are three cases of the relationship between d

_{xy}and d, which correspond to the three relationship cases of A and B. The three cases are elaborated below. Case (a) d

_{xy}= d. This case corresponds to the case of B = A. For such a case, one candidate cone is generated. Note that o

_{x}and o

_{y}should be on the diameter of the projection circle of the cone. Case (b) ${d}_{xy}$ > d. This case corresponds to the case of B > A. For such a case, two candidate cones taking $\stackrel{\rightharpoonup}{g{s}_{x}}$ and $\stackrel{\rightharpoonup}{g{s}_{y}}$ as directions are generated. Case (c) ${d}_{xy}$ < d. This case corresponds to the case of B < A. For such a case, four candidate cones are generated. For two of the cones, o

_{x}and o

_{y}should be on the arcs of the projection circles (at the top and at the bottom) of the cones. For another cone, o

_{x}should be on the projection circle (on the right), and the line $\overline{{o}_{x}{o}_{y}}$ should go through the center of the circle. For yet another cone, o

_{y}should be on the projection circle (on the left), and the line $\overline{{o}_{x}{o}_{y}}$ should go through the center of the circle. Note that for Case (c), the GCC algorithm generates four candidate cones to cover sensor nodes ${s}_{{a}_{x}}\text{and}{s}_{{a}_{y}}$. In practice, it greedily adjusts the cones to the extreme limitation with the purpose that the cones may at the same time cover as many as possible other sensor nodes.

_{i}. Corresponding to the power requirement, $EP\left(c,{s}_{i}\right)$ returns the power (in the unit of mW) received by s

_{i}from the energy emitted by charger c. As will be shown in Section 5, the received power can be calculated by interpolating power regression expressions derived from practical experimental data.

## 5. Analysis

#### 5.1. Algorithm Time Complexity Analysis

^{2}= O(mn

^{2}) and the time complexity to generate the candidate cones is also O(mn

^{2}). On each iteration in the repeat-until loop (Lines 18–24), the GCC algorithm first selects the cone c from CS that covers most unmarked nodes in NS (Line 19). Since there are at most mn

^{2}cones in CS, and the algorithm should check if a cone can cover all nodes, the selection can be done in O(mn

^{3}) time complexity. The GCC algorithm then checks if every unmarked node covered by cone c can be marked after updating its energy requirement (Lines 21–23). The checking takes O(n) time complexity. A cone can be selected only once in the repeat-until loop, so there are at most O(mn

^{2}) iterations in the loop. In summary, there are O(mn

^{2}) iterations in the repeat-until loop, and each iteration’s time complexity is O(mn

^{3}) (Line 19). Thus, the time complexity of the loop is O(m

^{2}n

^{5}). Since the repeat-until loop has the highest time complexity in the algorithm, we thus have that the time complexity of the GCC algorithm is O (m

^{2}n

^{5}).

^{2}= O(n

^{2}) adjustment tests. Since there are m grid points, the ACC algorithm forms m half spheres. Let the half spheres cover ${k}_{1},{k}_{2},\dots ,{k}_{m}$ sensor nodes, respectively. Thus, the total number of candidate cones is ${\sum}_{i=1}^{m}{k}_{i}$ ≤ mn = O(mn) and the total number of adjustment tests is of the time complexity O(mn

^{3}). On each iteration in the repeat-until loop (Lines 16–22), the ACC algorithm first selects the cone c from CS that covers the most unmarked nodes in NS (Line 17). Since there are at most mn cones in CS, and the algorithm should check if a cone can cover all nodes, the selection (Line 17) can be done in O(mn

^{2}) time complexity. The ACC algorithm then checks if every unmarked node covered by cone c can be marked after updating its energy requirement (Lines 19–21). The checking takes O(n) time complexity. A cone can be selected only once in the repeat-until loop, so there are at most O(mn) iterations in the loop. Thus, the time complexity of the repeat-until loop is O(m

^{2}n

^{3}). Since the repeat-until loop has the highest time complexity in the ACC algorithm, the time complexity of the ACC algorithm is O(m

^{2}n

^{3}).

#### 5.2. Grid Point Separation Analysis

^{′}within the area of C, we have $\overline{v{u}^{\prime}}\le D$. So, if there is at least one grid point $g$ within C, the sensor node located at v can be covered by a charger deployed at $g$. When Z = 0, v is on the floor plane and the circle C has the smallest radius $r=\sqrt{{D}^{2}-{H}^{2}}$. As shown in Figure 11, if $S>\sqrt{2}r$, then there may be no grid point within the area of circle C. On the contrary, if $S\le \sqrt{2}r$=$\sqrt{2({D}^{2}-{H}^{2})}$, there is at least one grid point within the area of circle C. In summary, In summary, S ≤$\text{}\sqrt{2({D}^{2}-{H}^{2})}$ ensures that every sensor node is covered by at least one charger deployed at one of the grid points.

^{′}be the circle centered at $g$ with radius r and let $a$ and b be the intersection of C and C

^{′}(Figure 12). Since C and C

^{′}have the same radius, we have that the angel $\angle agb$ associated with the minor arc $\widehat{ab}$ should be larger than 120°; otherwise, $g$ will not be within C. For neighboring grid points $g$, ${g}^{\prime}$ and ${g}^{\u2033}$ that are apart from each other with separation S, their associated angle $\angle {g}^{\prime}g{g}^{\u2033}$ is 90°. We have that either ${g}^{\prime}$ or ${g}^{\u2033}$ is within the sector associated with the minor arc $\widehat{ab}$. Therefore, there are at least two grid points within the area of C. That is to say, there are at least two chargers covering a sensor node. In summary, S ≤ $\sqrt{({D}^{2}-{H}^{2})}$ ensures that every sensor node is covered by at least two chargers deployed at two of the grid points.

_{GS}, the number of grid squares in circle C:

_{GP}, the number of grid points in circle C:

## 6. Experiments and Simulations

#### 6.1. Experiments

^{−2.217}, y = 3.8881x

^{−2.225}, y = 2.6394x

^{−1.821}, y = 1.3522x

^{−1.939}, y = 0.7134x

^{−2.01}, fit well with the experimental data, as they all have coefficient of determination (i.e., R

^{2}) [26] values larger than 0.958, which is very close to the maximum value 1. They also comply with the Friis equation, since the function exponents are approximately-2. Therefore, we may well estimate the power received by the harvester with the functions. In practice, we interpolate the values of the functions to estimate the received power for given parameters, like distance D, horizontal angle ф, and vertical angle θ.

#### 6.2. Simulations

^{2}) of candidate cones, while ACC has O(mn) of candidate cones, where n is the number of sensor nodes and m is the number of grid points.

^{2}n

^{3}) time complexity, while GCC has O(m

^{2}n

^{5}) time complexity. However, the time complexity of the big O notation is just an upper bound. Some experimental cases may have sensor nodes whose coverage requirements are met very early. For such cases, the algorithm execution may not be too long. This accounts for why the execution times of algorithms do not grow according to the functions of m

^{2}n

^{3}or m

^{2}n

^{5}.

^{2}n

^{3}) time complexity, while GCC has O(m

^{2}n

^{5}) time complexity.

## 7. Conclusions

^{2}n

^{3}) time complexity and GCC has O(m

^{2}n

^{5}) time complexity, where n is the number of sensor nodes and m is the number of grid points.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**A wireless rechargeable sensor network (WRSN) sensor node equipped with an energy harvester to harvest energy. RF: radio frequency.

**Figure 6.**The projection of a cone (respectively, vector) onto a unit sphere surface is a circle (respectively, point).

**Figure 10.**Illustration of the intersection of the grid plane with an upper half sphere of radius D (the charging distance) centered at a sensor node.

**Figure 15.**Experimental settings for (

**a**,

**b**) different horizontal angles (azimuths); and (

**c**,

**d**) different vertical angles (elevations).

**Figure 17.**The functions derived by performing power regression analysis on experimental data (ED) of the power received (PR) for horizontal angel ф = 0°, 15° and 30°, and vertical angle θ = 0°.

**Figure 18.**The functions derived by performing power regression analysis on ED of the PR for horizontal angel ф = 0°, and vertical angle θ = 0°, 15° and 30°.

**Figure 20.**The comparisons of GCC and ACC in terms of the number of deployed chargers for: (

**a**) 1-coverage; (

**b**) 2-coverage; and (

**c**) 3-coverage requirement.

**Figure 21.**The comparisons of GCC and ACC in terms of the execution time for: (

**a**) 1-coverage; (

**b**) 2-covearge; and (

**c**) 3-coverage requirements.

**Figure 22.**The comparisons of GCC and ACC in terms of the number of deployed chargers for: (

**a**) low; (

**b**) medium; and (

**c**) high power requirement.

**Figure 23.**The comparisons of GCC and ACC in terms of the execution time for: (

**a**) low; (

**b**) medium; and (

**c**) high power requirement.

D\ф | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
---|---|---|---|---|---|---|---|

0.5 m | 17.63 | 16.38 | 13.89 | 9.89 | 3.96 | 2.15 | 0.37 |

1.0 m | 6.3 | 5.72 | 3.96 | 2.29 | 1.14 | 0.58 | - |

1.5 m | 1.93 | 1.26 | 1 | 0.72 | 0.42 | - | - |

2.0 m | 1.39 | 1.04 | 0.8 | 0.57 | 0.25 | - | - |

2.5 m | 0.84 | 0.74 | 0.62 | 0.34 | - | - | - |

3.0 m | 0.47 | 0.31 | 0.18 | - | - | - | - |

3.5 m | 0.28 | 0.15 | - | - | - | - | - |

4.0 m | 0.21 | 0.11 | - | - | - | - | - |

4.5 m | 0.14 | - | - | - | - | - | - |

D\θ | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
---|---|---|---|---|---|---|---|

0.5 m | 17.63 | 13.52 | 9.07 | 5.55 | 2.78 | 0.92 | 0.26 |

1.0 m | 6.3 | 5.23 | 2.65 | 1.26 | 0.79 | 0.41 | - |

1.5 m | 1.93 | 1.52 | 1.02 | 0.56 | 0.29 | - | - |

2.0 m | 1.39 | 0.95 | 0.87 | 0.34 | 0.18 | - | - |

2.5 m | 0.84 | 0.61 | 0.76 | 0.28 | - | - | - |

3.0 m | 0.47 | 0.42 | 0.36 | 0.15 | - | - | - |

3.5 m | 0.28 | 0.21 | 0.19 | - | - | - | - |

4.0 m | 0.21 | 0.14 | - | - | - | - | - |

4.5 m | 0.14 | 0.12 | - | - | - | - | - |

Item | Parameter |
---|---|

Sensor deployment cuboid | 20 m (L) × 15 m (W) × 2.3 m (H) |

Number of sensors | $50,\text{}100,\text{}150,\text{}200,\text{}250$ |

Effective charging distance | 3 m (D) |

Opening angle | 30° (A) |

Height of grid points | 2.3 m (H) |

Grid point separation | 1.8 m (S) |

Coverage requirement of sensor nodes | 1-, 2-, or 3-coverage |

Energy requirement of sensor nodes | 0.18, 0.54, 0.9 mW |

Times of simulation per case | 30 times/case |

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**MDPI and ACS Style**

Jiang, J.-R.; Liao, J.-H.
Efficient Wireless Charger Deployment for Wireless Rechargeable Sensor Networks. *Energies* **2016**, *9*, 696.
https://doi.org/10.3390/en9090696

**AMA Style**

Jiang J-R, Liao J-H.
Efficient Wireless Charger Deployment for Wireless Rechargeable Sensor Networks. *Energies*. 2016; 9(9):696.
https://doi.org/10.3390/en9090696

**Chicago/Turabian Style**

Jiang, Jehn-Ruey, and Ji-Hau Liao.
2016. "Efficient Wireless Charger Deployment for Wireless Rechargeable Sensor Networks" *Energies* 9, no. 9: 696.
https://doi.org/10.3390/en9090696